Energy-preserving Pseudo-spectral Methods for Klein-Gordon-Schrödinger Equations

Transcription

1 .. Energy-preserving Pseudo-spectral Methods for Klein-Gordon-Schrödinger Equations Jingjing Zhang Henan Polytechnic University December 9, 211

2 Outline.1 Background.2 Aim, Methodology And Motivation.3 Construction of Conserved Schemes.4 Numerical Experiments.5 Conclusions

3 Background Outline.1 Background.2 Aim, Methodology And Motivation.3 Construction of Conserved Schemes.4 Numerical Experiments.5 Conclusions

4 Background Consider the coupled Klein-Gordon-Schrödinger (KGS) equations { iψ t ψ xx + ψφ =, φ tt φ xx + φ ψ 2 x R, t, i = 1, (1.1) =, with initial and boundary conditions ψ(x, ) = ψ (x), φ(x, ) = φ (x), φ t (x, ) = φ 1 (x), (1.2) lim x ψ(x, t) =, lim x φ(x, t) =, (1.3) where ψ(x, t) is complex representing a scalar neutron field, and φ(x, t) is real representing a scalar neutral meson field, respectively. The KGS equation (1.1) describes the interaction between conservative complex neutron field and neutral meson Yukawa in quantum field theory (I. Fukuda, M. Tsutsumi, 1975,1978).

5 Background Existing References: Theoretical research: I. Fukuda, M. Tsutsumi(1975,1978); B. Guo(1982), B. Guo, C. Miao(1985); M. Wang, Y. Zhou(23)... Numerical research: Xiang(1988): Conservative spectral method and error estimate without simulation Zhang(25): Conservative difference scheme without simulation Kong, Liu, Xu(26) Multisymplectic discretization Bao, Yang(27): Fourier pseudo-spectral in space+ appropriate transmission condition or CN/LP for time derivatives Hong, Jiang, Kong, Li(27):Five difference schemes comparison Kong, Hong, Liu(28): Difference operators + symplectic Hong, Jiang, Li(29): Explicit multi-symplectic methods Kong, Zhang, et al.(21): Fourier pseudo-spectral +SV...

6 Background Let ψ(x, t) = p(x, t) + iq(x, t), φ t (x, t) = 2v(x, t), where both p(x, t) and q(x, t) are real-valued functions. Let z = [q, v, p, φ] T, then KGS eq. can be written into the infinite dimensional Hamiltonian form: δh(z) z t = J 4, where J 4 = δz [ 2 J 2 J 2 2 ], J 2 = [ 1 1 The Hamiltonian function H(z) = [ 1 2 φ(p2 + q 2 ) 1 4 (φ2 + φ 2 x + px 2 + qx) 2 v 2 ]dx. R R ]. (1.4) is an invariant of KGS. The charge is also an invariant of KGS: Q(t) = ψ 2 = ψ 2 dx = ψ (x) 2 dx = Q(). (1.5) R

7 Aim, Methodology And Motivation Outline.1 Background.2 Aim, Methodology And Motivation.3 Construction of Conserved Schemes.4 Numerical Experiments.5 Conclusions

8 Aim, Methodology And Motivation Aim construct schemes to preserve the conserved quantity of KGS. Methodology Sinc Pseudo-spectral discretization in space + discrete gradient integrator in time Motivation 1. Sinc: unbounded domain+symmetric 2nd differential matrix 2. Discrete gradient method is a unified IPI ( G. R. W. Quispel and G. S. Turner, J. Phys. A. 1996)

9 Aim, Methodology And Motivation Aim construct schemes to preserve the conserved quantity of KGS. Methodology Sinc Pseudo-spectral discretization in space + discrete gradient integrator in time Motivation 1. Sinc: unbounded domain+symmetric 2nd differential matrix 2. Discrete gradient method is a unified IPI ( G. R. W. Quispel and G. S. Turner, J. Phys. A. 1996)

10 Aim, Methodology And Motivation Aim construct schemes to preserve the conserved quantity of KGS. Methodology Sinc Pseudo-spectral discretization in space + discrete gradient integrator in time Motivation 1. Sinc: unbounded domain+symmetric 2nd differential matrix 2. Discrete gradient method is a unified IPI ( G. R. W. Quispel and G. S. Turner, J. Phys. A. 1996)

11 Aim, Methodology And Motivation Aim construct schemes to preserve the conserved quantity of KGS. Methodology Sinc Pseudo-spectral discretization in space + discrete gradient integrator in time Motivation 1. Sinc: unbounded domain+symmetric 2nd differential matrix 2. Discrete gradient method is a unified IPI ( G. R. W. Quispel and G. S. Turner, J. Phys. A. 1996)

12 Aim, Methodology And Motivation Introduction to Sinc Pseudo-spectral Method(F. Stenger, JCAM, 2) Sinc pseudo-spectral method in spatial direction is conducted as follows: Step 1 Denoting h = b a N 1, the space variable is discretized into the collocation, x 1 = a,..., x i = a + (i 1)h,..., x N = b. Interpolating and approximating the unknown function u(x, t) by Sinc function, N u(x, t) u N (x, t) = S i (x; h)u i (t), where u i (t) = u(x i, t) denotes the values of u in the nodal points of the chosen collocation and S i (x; h) = sinz i z i, z i = π h (x a (i 1)h), S i (x j ) = δ ij. i=1

13 Aim, Methodology And Motivation Introduction to Sinc Pseudo-spectral Method(Cont.) Step 2 Under the collocation-interpolation defined in step 1, the second order spatial derivative of u in the nodal point of collocation is approximated by where 2 u x 2 (x j, t) 2 u N x 2 (x j, t) = N b ij u i (t), i=1 b ii = 1 3 (π h )2, b ij = 2( 1)j i+1 h 2 (j i) 2. (2.1) Let B = (b ij ) and B is a symmetric matrix representing the spectral differential matrix corresponding to xx.

14 Aim, Methodology And Motivation Introduction to Discrete Gradient Method. Definition 1. A differential function I (y) is called a first integral of differential equations di (y). ẏ = f (y) if dt = I (y)f (y) =.. Definition 2 (Quispel, Turner(1996)). Let I (y, ȳ) be a continuous function of y and ȳ, is called a discrete gradient of I (y) if it satisfies I (y, ȳ) T (ȳ y) = I (ȳ) I (y) and I (y, y) = I (y). Furthermore, if I (y, ȳ) = I (ȳ, y) holds, we call it. the symmetric discrete gradient.

15 Aim, Methodology And Motivation Introduction to Discrete Gradient Method(Cont.) Examples of discrete gradient: Coordinate increment discrete gradient 1 I (y, ȳ): 1 I (y, ȳ) := I (ȳ 1,y 2,y 3,...,y d ) I (y 1,y 2,y 3,...,y d ) ȳ 1 y 1 I (ȳ 1,ȳ 2,y 3,...,y d ) I (ȳ 1,y 2,y 3,...,y d ) ȳ 2 y 2. I (ȳ 1,ȳ 2,ȳ 3,...,ȳ d ) I (ȳ 1,ȳ 2,...,ȳ d 1,y d ) ȳ d y d. (2.2) Symmetric coordinate increment discrete gradient I 2 (y, ȳ): I 2 (y, ȳ) = 1 2 ( 1 I (y, ȳ) + 1 I (ȳ, y)). (2.3)

16 Construction of Conserved Schemes Outline.1 Background.2 Aim, Methodology And Motivation.3 Construction of Conserved Schemes.4 Numerical Experiments.5 Conclusions

17 Construction of Conserved Schemes The construction of conserved schemes are conducted in two steps: Firstly we apply the Sinc pseudo-spectral method in spatial direction to the KGS and reduce the infinite-dimensional Hamiltonian system (1.4) to the finitedimensional one Q t = 1 2BP + Φ P, V t = 1 2 BΦ 1 2 Φ (P 2 +Q 2), P t = ( 1 (3.1) 2BQ + Φ Q), Φ t = ( 2V ), where B = (b ij ) is defined by (2.1), and Q = [q 1, q 2,, q N ] T, P = [p 1, p 2,, p N ] T, V = [v 1, v 2,, v N ] T and Φ = [φ 1, φ 2,, φ N ] T, where denotes component multiplication between two vectors with the same length.

18 Construction of Conserved Schemes Secondly, since the Hamiltonian H(Q, V, P, Φ) = 1 2 N j=1 φ j(p 2 j + q 2 j ) (PT BP + Q T BQ + Φ T BΦ Φ T Φ) V T V is a first integral of (3.1), by using the coordinate increment discrete gradient we obtain the conserved numerical schemes I: Q n+1 = Q n + τ(φ n Pn+1 + P n B Pn+1 + P n ) (3.2a) 2 V n+1 = V n + τ( 1 2 B Φn+1 + Φ n 1 Φ n+1 + Φ n (Pn+1 2 +Q n+1 2)) (3.2b) P n+1 = P n τ(φ n Qn+1 + Q n 2 Φ n+1 = Φ n + τ(v n+1 + V n ) B Qn+1 + Q n ) (3.2c) 2 (3.2d) where τ is the time step size and [Q nt, V nt, P nt, Φ nt ] T is the discretization at time t = nτ.

19 Construction of Conserved Schemes By using the symmetric coordinate increment discrete gradient, we obtain the conserved numerical schemes II: Q n+1 = Q n + τ( 1 4 (Φn+1 + Φ n ) (P n+1 + P n ) B Pn+1 + P n ) 2 (3.3a) V n+1 = V n + τ( 1 2 B Φn+1 + Φ n 1 Φ n+1 + Φ n (3.3b) (Pn+1 2 +Q n+1 2 +P n 2 +Q n 2)) P n+1 = P n τ( 1 4 (Φn+1 + Φ n ) (Q n+1 + Q n ) B Qn+1 + Q n ) (3.3c) 2 Φ n+1 = Φ n + τ(v n+1 + V n ) (3.3d) where τ is the time step size and [Q nt, V nt, P nt, Φ nt ] T is the discretization at time t = nτ.

20 Construction of Conserved Schemes. Theorem 3. The scheme I (3.2a)-(3.2d) has the following properties: (1) i Ψn+1 Ψ n + 1 τ 4 B(Ψn+1 + Ψ n ) + Φ n Ψn+1 + Ψ n =, 2 Φ n+1 2Φ n + Φ n 1 τ (B I N)(Φ n+1 + 2Φ n + Φ n 1 ) 1 2 ( Ψn Ψ n 2) =. (2) Q N = Q N 1 = = Q, H N = H N 1 = = H, where Q n = h N ψj n 2, H n = 1 N 2 j=1 φn j (pn 2 j + qj n2 ) (< Pn, BP n > j=1 + < Q n, BQ n > + < Φ n, BΦ n > < Φ n, Φ n >) < V n, V n >. (3)It has first order accuracy in the time direction and can be implemented explicitly..

21 Construction of Conserved Schemes. Theorem 4. The scheme II (3.3a)-(3.3d) has the following properties: (1). i Ψn+1 Ψ n + 1 τ 4 B(Ψn+1 + Ψ n ) (Φn+1 + Φ n ) (Ψ n+1 + Ψ n ) =, Φ n+1 2Φ n + Φ n 1 τ (B I N)(Φ n+1 + 2Φ n + Φ n 1 ) 1 4 ( Ψn Ψ n 2 + Ψ n 1 2) =. (2) It preserves the charge Q n and energy H n exactly. (3) It has second order accuracy in the time direction.

22 Numerical Experiments Outline.1 Background.2 Aim, Methodology And Motivation.3 Construction of Conserved Schemes.4 Numerical Experiments.5 Conclusions

23 Numerical Experiments Example 1 Consider KGS (1.1) with analytic solitary solutions [wang, zhou(23)]: { ψ(x, t, v, x ) = v 2 sech2 1 2 (x vt x 1 v ) exp(i(vx + 1 v 2 +v 4 2 2(1 v 2 ) t)) φ(x, t, v, x ) = 3 4(1 v 2 ) sech2 1 2 (x vt x 1 v ), 2 where v < 1 is the propagating velocity of the wave and x is the initial phase. Define L 2 and L norms as follows: (4.1) e Ψ 2 2 = h j e Φ 2 2 = h j Ψ n j Ψ(x j, t n ) 2, e Ψ = max Ψ n j Ψ(x j, t n ), j Φ n j Φ(x j, t n ) 2, e Φ = max Φ n j Φ(x j, t n ). j

24 Numerical Experiments Table 1: Convergence rate in the time direction on [ 2, 2] [, 6]. τ e Ψ 2.446e e e e e-4 order e Φ e e e e e-4 order e Ψ e e e e e-4 order e Φ e e e e e-4 order where order ln( e u(τ) p / e u ( τ 2 ) p) ln(2), e u means e Ψ or e Φ and p represents L or L 2 norm.(n = 11, v =.3, x = 5)

25 Numerical Experiments Table 2: Convergence rate in the time direction on [ 2, 2] [, 6]. τ e Ψ e e e e e-5 order e Φ e e e e e-6 order e Ψ e e e e e-5 order e Φ e e e e e-6 order where order ln( e u(τ) p / e u ( τ 2 ) p) ln(2), e u means e Ψ or e Φ and p represents L or L 2 norm. (N = 11, v =.5, x = 5)

26 Numerical Experiments 4 x 1 13 x 1 13 Error of Charge 2 2 Error of Energy x x 1 15 Error of Charge 5 Error of Energy Figure 1: Errors of charge and energy of conserved scheme 1(the top two) and 2(the below two)([ 4, 4] [, 1], N = 11, τ =.2, a =.5, x = 25).

27 Numerical Experiments Example 2 Symmetric soliton-soliton collision with initial values: ψ (x) = ψ(x, t, v 1, x 1 ) t= + ψ(x, t, v 2, x 2 ) t=, φ (x) = φ(x, t, v 1, x 1 ) t= + φ(x, t, v 2, x 2 ) t=, φ 1 (x) = t φ(x, t, v 1, x 1 ) t= + t φ(x, t, v 2, x 2 ) t=, (4.2) where ψ(x, t, v i, x i ), φ(x, t, v i, x i ), i = 1, 2 is defined by (4.1) and v 1 =.4, x 1 = 2, v 2 =.4, x 2 = Ψ 1 4 Φ x t x t Figure 2: Evolution of Ψ and Φ of conserved scheme 1.

28 Numerical Experiments 2 x x Error of Charge Error of Energy Figure 3: Errors of charge and energy of conserved scheme 1([ 4, 4] [, 8], N = 11, τ =.5, v 1 =.4, x 1 = 2, v 2 =.4, x 2 = 2, T = 8).

29 Numerical Experiments Ψ Φ x t 6 2 x t 6 Figure 4: Evolution of Ψ and Φ of conserved scheme 2.

30 Numerical Experiments 1.5 x x Error of Charge.5.5 Error of Energy Figure 5: Errors of charge and energy of conserved scheme 2([ 4, 4] [, 8], N = 11, τ =.5, v 1 =.4, x 1 = 2, v 2 =.4, x 2 = 2, T = 8).

31 Numerical Experiments Example 3 Antisymmetric soliton-soliton collision with initial values: ψ (x) = ψ(x, t, v 1, x 1 ) t= + ψ(x, t, v 2, x 2 ) t=, φ (x) = φ(x, t, v 1, x 1 ) t= + φ(x, t, v 2, x 2 ) t=, φ 1 (x) = t φ(x, t, v 1, x 1 ) t= + t φ(x, t, v 2, x 2 ) t=, (4.3) where ψ(x, t, v i, x i ), φ(x, t, v i, x i ), i = 1, 2 is defined by (4.1) and v 1 =.6, x 1 = 2, v 2 =.8, x 2 = 3. Please see the movies of the solution evolution generated by scheme 1, scheme 2 and midpoint scheme.

32 Numerical Experiments x 1 14 x 1 14 Error of Charge Error of Energy Figure 6: Errors of charge and energy of conserved scheme 1([ 45, 45] [, 5], N = 21, τ =.1, v 1 =.5, x 1 = 2, v 2 =.8, x 2 = 3, T = 5).

33 Numerical Experiments 2 x x Error of Charge 1.5 Error of Energy Figure 7: Errors of charge and energy of conserved scheme 2([ 45, 45] [, 5], N = 21, τ =.1, v 1 =.5, x 1 = 2, v 2 =.8, x 2 = 3, T = 5).

34 Numerical Experiments Error of Charge x Error of Energy x Figure 8: Errors of charge and energy of midpoint scheme ([ 45, 45] [, 5], N = 21, τ =.1, v 1 =.5, x 1 = 2, v 2 =.8, x 2 = 3, T = 5).

35 Conclusions Outline.1 Background.2 Aim, Methodology And Motivation.3 Construction of Conserved Schemes.4 Numerical Experiments.5 Conclusions

36 Conclusions Object: KGS equation (1.1) Viewpoint: Infinite dimensional Hamiltonian System Aim: Construct conserved numerical schemes. Methodology: Sinc pseudospectral method in space + discrete gradient method in time. Advantages: S1 Easy implementation and good conservation properties.

37 Conclusions Thank you for your attention!